GKLO representations of twisted Yangians in type $\mathsf{A}\mathsf{I}$ and quantizations of symmetric quotients of the affine Grassmannian

We construct an analogue of Gerasimov-Kharchev-Lebedev-Oblezin (GKLO) representations for twisted Yangians of type $\mathsf{A}\mathsf{I}$, using the recently found current presentation of these algebras due to Lu, Wang and Zhang. These new representations allow us to define interesting truncations of twisted Yangians, which, in the spirit of Ciccoli-Drinfeld-Gavarini quantum duality, reflect the Poisson geometry of homogeneous spaces. As our main result, we prove that a truncated twisted Yangian quantizes a scheme supported on quotients of transverse slices in the affine Grassmannian.